Lecturer: |
S. Crépey |

Period: |
Term 3 |

ECTS: |
6 |

Hourly volume: |
3 hours per week |

This course deals with important regulatory quant modeling issues and with a panoply of numerical optimization problems that are the bread and butter of quants’ activity.

**Part I XVA Analysis **

Since 2008, investment banks compute various X-valuation adjustments (XVAs) to assess counterparty risk and its funding and capital implications. XVAs deeply affect the derivative pricing task by making it global (portfolio-wide), nonlinear, and entity dependent. A proper financial understanding of even the first generation XVAs (CVA, DVA, and FVA, where C sits for credit, D for debt, and F for funding) points out to the distinction between firm and shareholder valuation, which mathematically goes to enlargement of filtration and singular probability change. Second generation XVAs involve not only conditional expectations (i.e. prices), but also conditional risk measures (value-at-risk and expected shortfall) for MVA (margin valuation adjustment) and KVA (capital valuation adjustment) computations. The course will provide an up-to-date covering of the XVA universe and of the embedded risk measure issues, from the triple angle of finance (wealth transfers), stochastic analysis (enlargement of filtration and backward SDEs), and computations (nested Monte Carlo and neural net regressions).

*Outline*

1 The XVA cost-of-capital approach in a static setup

2 The XVA cost-of-capital approach in continuous time

3 XVA analysis of bilateral trade portfolios

4 XVA nested Monte Carlo computational strategies

5 XVA neural net regression and quantile regression computational strategies

6 XVA analysis of centrally cleared portfolios

**Part II Calibration, Training, and Other Optimization Problems**

The implementation of approaches and models is conditioned by the solution of a number of numerical optimization tasks, such as: calibration of pricing models; training of supervised and unsupervised learning models for proxy pricing, pricing and hedging, or anomaly detection; collateral and capital compression schemes.

*Outline*

0 Prerequisites in optimization: (possibly stochastic) gradient descents, handling constraints, derivative-free methods

1 The ill-posed inverse calibration problem

2 Extracting the yield curve from bond prices

3 Calibration of the risk-neutral density from option prices

4 Calibration of the local volatility, old and new (Dupire and Gatheral formulas, Tikhonov regularization, Gaussian processes vs. neural net metamodeling)

5 Proxy pricing by Gaussian process regression

6 Financial nowcasting (completion of tensors such as interest rate curves, volatility surfaces,..) and anomaly detection by auto-encoder compression / completion schemes

*Tutorial*s in python/tensorflow.

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**References****: **Related material on *HYPERLINK "https://www.lpsm.paris/pageperso/crepey/"*

**Keywords**: model calibration, XVA analysis (CVA, FVA, KVA), central counterparties (CCP).

**Prior knowledge**: Stochastic analysis, mathematical finance, numerical finance at MSc level. Some knowledge of corporate finance or programming skills in python and/or C++ are also useful but can be acquired “on the job”.

**Assessment**: Programming projects to be delivered as ** python jupyter notebooks **running under Google Colab